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Symbolic and Numerical Methods for Searching Symmetries of Ordinary Differential Equations with a Small Parameter and Reducing Its Order

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Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

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Abstract

Two programs for computer algebra systems are described that deal with Lie algebras of generators admitted by systems of ordinary differential equations (ODEs). The first one allows to find the generators of admitted transformations in the specified form. This program is written in Python and based on SciPy library. It does not require solving partial differential equations symbolically and can also analyze equations with Riemann–Liouville fractional derivatives and find approximate symmetries for systems of equations with a small parameter. The second program written as a package for Maple computes the operator of invariant differentiation in special form for given Lie algebra of generators. This operator is used for order reduction of given ODE systems.

Supported by the Ministry of Science and High Education of the Russian Federation (State task No. 1.3103.2017/4.6).

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Acknowledgments

We are grateful to Prof. R.K. Gazizov and Prof. S.Yu. Lukashchuk for constructive discussion. Also we thank the referees whose comments helped us a lot to improve the early draft of this paper.

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Authors and Affiliations

  1. Ufa State Aviation Technical University, Ufa, Russia

    Alexey A. Kasatkin & Aliya A. Gainetdinova

Authors
  1. Alexey A. Kasatkin
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  2. Aliya A. Gainetdinova
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Correspondence to Alexey A. Kasatkin .

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Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Kasatkin, A.A., Gainetdinova, A.A. (2019). Symbolic and Numerical Methods for Searching Symmetries of Ordinary Differential Equations with a Small Parameter and Reducing Its Order. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_19

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  • DOI: https://doi.org/10.1007/978-3-030-26831-2_19

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